Better-balanced-pair inequality conjecture for the cd-index

At least 23 years old · documented by 2002

Let $(m_1,n_1)$ and $(m_2,n_2)$ be pairs of non-negative integers, where $(m_1,n_1)$ is strictly better balanced than $(m_2,n_2)$ , meaning that the first pair is strictly closer to balance than the second in the sense used in the paper. Let $L$ , $M$ , and $N$ be lists. Better-balanced-pair inequality conjecture. Then

\beta(M,m_1,L,n_1,N)>\beta(M,m_2,L,n_2,N).

This extends the preceding pairwise comparison to arbitrary lists inserted between and around the two entries; the source gives no resolution.

References

Primary source

Swapneel Mahajan, “The cd-index of the Boolean lattice”, arXiv:math/0211390 (2002).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-written calculation claims a counterexample disproves the conjecture, but no independent verification has appeared.

Mahajan formulated this extension of the balance conjecture for Boolean-lattice $cd$ -coefficients and recorded it as Conjecture 4. It asserts that making the designated pair closer to balance strictly increases the coefficient, even with arbitrary surrounding lists.

Known results

Mahajan (2002): proved partial balance inequalities in Theorems 4 and 5.
Mahajan (2002): established related reverse-unimodality inequalities and characterized the maximum coefficient in each degree.
Ehrenborg and Readdy (2019): surveyed these results without resolving the arbitrary-list conjecture.

Posted attempt

A reader-written computation claims a counterexample in $cd$ -monomial degree $25$ , using $(m_1,n_1)=(5,5)$ and $(m_2,n_2)=(4,6)$ , with surrounding lists $(0)$ and $(0,7)$ . It reports the supposedly better-balanced case has the smaller coefficient, contradicting the conjectured strict inequality. The calculation has not been independently verified.

Current status (as of August 2026): An unverified complete counterexample claim is recorded, while Mahajan’s partial results are established and no independently verified proof or disproof is available.

Sources

CounterexampleThis solution needs a summarySee full solutionHide full solution

The conjecture is false for $cd$ -monomials of degree $25$ .

Use the list notation

(a_1,\ldots,a_r)=c^{a_1}dc^{a_2}d\cdots dc^{a_r},

and take

M=(0),\qquad L=(0,7),\qquad N=\varnothing,

together with

(m_1,n_1)=(5,5),\qquad(m_2,n_2)=(4,6).

Both pairs have sum $10$ , and

|5-5|=0<2=|4-6|,

so $(5,5)$ is strictly better balanced than $(4,6)$ . The conjecture therefore predicts

\beta(0,5,0,7,5)>\beta(0,4,0,7,6). \tag{1}

The coefficients can be computed directly from the established Lemmas 3.2 and 4.4 of the source. For every nonempty list $a=(a_1,\ldots,a_r)$ , these give

\begin{aligned} \beta(a) ={}&\sum_{\substack{1\leq i\leq r\\a_i>0}} \beta(a_1,\ldots,a_i-1,\ldots,a_r)\\ &+\sum_{i=1}^{r-1} \beta(a_1,\ldots,a_{i-1},a_i+a_{i+1}+1, a_{i+2},\ldots,a_r), \end{aligned} \tag{2}

with initial condition $\beta((0))=1$ . Every list on the right has degree one less than $a$ , where

\deg(a)=\sum_i a_i+2(r-1),

so (2) is a finite exact integer recursion.

Evaluating (2) gives

\beta(0,5,0,7,5)=1{,}287{,}894{,}857{,}340,

whereas

\beta(0,4,0,7,6)=1{,}288{,}645{,}806{,}880.

Thus

\beta(0,5,0,7,5)-\beta(0,4,0,7,6) =-750{,}949{,}540<0,

which is the opposite of (1).

As an independent coefficient recurrence, Proposition 2 gives

\Psi(B_{r+1})=\Psi(B_r)c+D(\Psi(B_r)), \qquad D(c)=d,\quad D(d)=cd,

with $D$ a derivation. Therefore $b(\varnothing)=1$ and

b(w) =\mathbf1_{\{w\text{ ends in }c\}}b(w\text{ with its last }c\text{ removed}) +\sum_{\text{$d$-positions}}b(w\text{ with that }d\text{ replaced by }c) +\sum_{\text{$cd$-substrings}}b(w\text{ with that }cd\text{ replaced by }d).

This separate recursion yields the same two values. Hence the more balanced pair gives a strictly smaller Boolean-lattice $cd$ -coefficient, disproving the conjecture.